The impact of heterogeneous dependency strength on the robustness of multi-layer networks

The representation of real-world complex systems using multi-layer networks has attracted significant research attention. However, the exploration of multi-layer networks with heterogeneous dependency strengths, reflecting node properties in complex systems more accurately, remains under explored. In this study, we extend the homogeneous dependency model to introduce degree-dependent heterogeneous dependency strengths: degree-positive and degree-negative. This establishes a theoretical framework for multi-layer networks with heterogeneous dependency strengths, which are based on node attributes. We derive accurate equations for simulating percolation processes in these networks and confirm the results through computational experiments. Comparative analysis of percolation phase transitions and the resilience of dependent layers under random attacks reveals distinct characteristics of this model, enhancing understanding of real-world complex systems. Our findings provide theoretical insights for designing resilient multi-layer networks with heterogeneous dependency strengths, better reflecting the complexities of real-world systems.